This Demonstration shows that the area under the first hump of a cycloid is three times the area of its generating circle. When you slide the "roll" slider, slices form a circle of radius two times the radius of the generating circle and a circular hole as large as the generating circle. As the number of slices goes to infinity, the figure approaches a perfect cycloid (as a region) and circles. Therefore, the difference between four times the larger disk and a hole of the same size gives the area under the cycloid.